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Search: id:A122945
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| A122945 |
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Recursive polynomias (p(k, x) = p(k - 1, x) - x^2*p(k - 2, x) ) used to produce a set of matrices a(i,j) at level n that then produce the characteristic polynomials which provide the triangular sequence t(n,m). |
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+0
3
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| 1, 1, -1, -1, 1, 1, 1, -1, 0, -1, -1, 1, 1, -2, 1, 1, -1, -2, 3, -1, -1, -1, 1, 3, -4, 0, 3, 1, 1, -1, -4, 5, 2, -6, 2, -1, -1, 1, 5, -6, -5, 10, -2, -4, 1, 1, -1, -6, 7, 9, -15, 0, 10, -3, -1, -1, 1, 7, -8, -14, 21, 5, -20, 5, 5, 1, 1, -1, -8, 9, 20, -28, -14, 35, -5, -15, 4, -1, -1, 1, 9, -10, -27, 36, 28, -56, 0, 35, -9, -6, 1, 1, -1, -10, 11
(list; table; graph; listen)
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OFFSET
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1,14
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COMMENT
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It was a real problem getting the matrices to agree with the polynomials: I was getting shift function polynomials!) 1 X 1 {{1}} 2 X 2 {{0, 1}, {1, -1}} 3 X 3 {{0, 1, 0}, {0, 0, 1}, {1, -1, 0}} 4 X 4 {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, -1, -1, 2}} 5 X 5 {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, -1, -2, 3, -1}}
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FORMULA
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p(k, x) = p(k - 1, x) - x^2*p(k - 2, x) p(k,n)->t0(i,j) t0(i,j)->a[i,j) a(i,j)->p'(n,x) p'(n,k)->t(n,m)
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EXAMPLE
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Input triangular sequence from the recurvise polynomials:
{{1},
{-1,1},
{-1, 1, -1},
{-1, 1, 0, -1},
{-1, 1, 1, -2, 1},
{-1, 1, 2, -3, 1, 1}
Output triangular sequence from characteristic polynomials of matrices:
{1},
{1, -1},
{-1, 1, 1},
{1, -1, 0, -1},
{-1, 1, 1, -2, 1},
{1, -1, -2, 3, -1, -1}
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MATHEMATICA
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p[0, x] = 1; p[1, x] = x - 1; p[k_, x_] := p[k, x] = p[k - 1, x] - x^2*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 20}] ; An[d_] := Table[If[n == d, -w[[n]][[m]], If[m == n, 1, 0]], {n, 2, d}, {m, 1, d - 1}]; Table[An[d], {d, 2, 19}] b = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], \ {d, 2, 19}]]; Flatten[%]
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CROSSREFS
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Sequence in context: A122520 A058393 A131256 this_sequence A119338 A054124 A096670
Adjacent sequences: A122942 A122943 A122944 this_sequence A122946 A122947 A122948
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 24 2006
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